2 HOL/Multivariate_Analysis/
3 ######################### _multi_variate ... nothing else found
5 src$ grep -r "interior " *
6 ==========================
7 HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
8 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
9 definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
11 grep -r "definition \"interval" *
12 =================================
13 HOL/Multivariate_Analysis/Brouwer_Fixpoint.thy
14 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
15 definition "interval_bij = (\<lambda> (a::'a,b::'a) (u::'a,v::'a) (x::'a::ordered_euclidean_space).
16 (\<chi>\<chi> i. u$$i + (x$$i - a$$i) / (b$$i - a$$i) * (v$$i - u$$i))::'a)"
19 ??? "{a<..<b} \<subseteq> {c..d} \<union> s" ?definition interval?
21 src$ grep -r ".nti.eriv" *
22 =========================
24 # HOL/Multivariate_Analysis/Derivative.thy
25 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
26 header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
27 definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
28 "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
30 definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
31 "f differentiable_on s \<equiv> (\<forall>x\<in>s. f differentiable (at x within s))"
33 definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"
36 =========================
37 HOL/Multivariate_Analysis/Integration.thy
38 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
39 header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
40 definition "integral i f \<equiv> SOME y. (f has_integral y) i"
43 =========================
44 HOL/Multivariate_Analysis/Real_Integration.thy
45 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
46 text{*We follow John Harrison in formalizing the Gauge integral.*}
48 definition Integral :: "real set \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool" where
49 "Integral s f k = (f o dest_vec1 has_integral k) (vec1 ` s)"
51 Multivariate_Analysis/L2_Norm.thy:header {* Square root of sum of squares *}
54 ################################################################################
56 ################################################################################
57 src/HOL$ grep -r " sum " *
58 ==========================
59 ex/Summation.thy:text {* The formal sum operator. *}
60 ex/Termination.thy:function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat"
61 ex/Termination.thy: "sum i N = (if i > N then 0 else i + sum (Suc i) N)"
62 Isar_Examples/Summation.thy
63 ~~~~~~~~~~~~~~~~~~~~~~~~~~~
64 text {* The sum of natural numbers $0 + \cdots + n$ equals $n \times
68 header{*Finite Summation and Infinite Series*}
73 deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
74 --{*Differentiation: D is derivative of function f at x*}
75 ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
76 "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"